Speed Prior-Based Inductive Inference

Given , as we observe an initial segment of some string,
which is the most likely continuation?
According to Bayes,

(5)

where
is the measure of , given . Having
observed we will predict those that maximize .
Which are those? In what follows, we will confirm the intuition that
for
the only probable continuations of
are those with fast programs. The sheer number of ``slowly'' computable
strings cannot balance the speed advantage of ``more quickly'' computable
strings with equal beginnings.

Proof.
Since no program that requires at least steps for producing
can compute in a PHASE with number , we have

Here we have used the Kraft inequality [13]
to obtain a rough upper bound for the enumerator:
when no is prefix of another one, then
Q.E.D.

Hence, if we know a rather fast finite program for , then Theorem
1 allows for predicting: if we observe some (
sufficiently large) then it is very unlikely that it was produced by an
-computing algorithm much slower than .

Among the fastest algorithms for is FAST itself, which is at
least as fast as , save for a constant factor. It outputs after
steps.